{"data":{"id":"10.4231/r7j9649p","type":"dois","attributes":{"doi":"10.4231/r7j9649p","prefix":"10.4231","suffix":"r7j9649p","identifiers":[],"alternateIdentifiers":[],"creators":[{"name":"Dong, Suchuan","givenName":"Suchuan","familyName":"Dong","affiliation":["Purdue University"],"nameIdentifiers":[{"schemeUri":"https://orcid.org","nameIdentifier":"https://orcid.org/0000-0001-6778-0679","nameIdentifierScheme":"ORCID"}]}],"titles":[{"title":"Incompressible Multiphase Flows: Physical Formulation and Numerical Algorithm"}],"publisher":"Purdue University Research Repository","container":{},"publicationYear":2015,"subjects":[{"subject":"Mathematics"},{"subject":"FOS: Mathematics","schemeUri":"http://www.oecd.org/science/inno/38235147.pdf","subjectScheme":"Fields of Science and Technology (FOS)"},{"subject":"N-phase flow"},{"subject":"phase field"},{"subject":"multiphase flows"},{"subject":"surface tension"},{"subject":"spectral element"},{"subject":"general order parameters"},{"subject":"pairwise surface tension"},{"subject":"applied mathematics"},{"subject":"computational physics"},{"subject":"Physics"}],"contributors":[],"dates":[{"date":"2015-04-07","dateType":"Available"},{"date":"2015-04-07","dateType":"Submitted"},{"date":"2015-04-07","dateType":"Accepted"},{"date":"2015","dateType":"Issued"}],"language":"en","types":{"ris":"DATA","bibtex":"misc","citeproc":"dataset","schemaOrg":"Dataset","resourceType":"Datasets","resourceTypeGeneral":"Dataset"},"relatedIdentifiers":[{"relationType":"References","relatedIdentifier":"10.1016/j.jcp.2014.11.039","resourceTypeGeneral":"JournalArticle","relatedIdentifierType":"DOI"}],"relatedItems":[],"sizes":["18 files","9 MB"],"formats":["image/jpeg","application/pdf"],"version":"1.0","rightsList":[{"rights":"Creative Commons Attribution Non Commercial No Derivatives 3.0 Unported","rightsUri":"https://creativecommons.org/licenses/by-nc-nd/3.0/legalcode","schemeUri":"https://spdx.org/licenses/","rightsIdentifier":"cc-by-nc-nd-3.0","rightsIdentifierScheme":"SPDX"}],"descriptions":[{"description":"\u0026lt;p\u0026gt;We present a family of physical formulations,\u0026amp;nbsp;and a\u0026amp;nbsp; numerical algorithm,\u0026amp;nbsp;based on a class of general order parameters for simulating\u0026amp;nbsp;the motion of a mixture of N\u0026amp;nbsp;(N \u0026amp;gt;=\u0026amp;nbsp;2)\u0026amp;nbsp;immiscible incompressible fluids\u0026amp;nbsp;with given densities, dynamic viscosities, and pairwise\u0026amp;nbsp;surface tensions.\u0026amp;nbsp;The N-phase formulations stem from a phase field model\u0026amp;nbsp;we developed in a recent work based on\u0026amp;nbsp;the conservations of mass/momentum, and the\u0026amp;nbsp;second law of thermodynamics.\u0026amp;nbsp;The introduction of general order\u0026amp;nbsp;parameters leads to an extremely strongly-coupled system\u0026amp;nbsp;of (N-1)\u0026amp;nbsp;phase field equations.\u0026amp;nbsp;On the other hand, the general form enables one to compute the N-phase\u0026amp;nbsp;mixing energy density coefficients in\u0026amp;nbsp;an explicit\u0026amp;nbsp;fashion in terms of\u0026amp;nbsp;the pairwise surface tensions.\u0026amp;nbsp;We show that the increased complexity in the form of\u0026amp;nbsp;the\u0026amp;nbsp;phase field equations\u0026amp;nbsp;associated with general order parameters\u0026amp;nbsp;in actuality\u0026amp;nbsp;does not cause\u0026amp;nbsp;essential computational difficulties.\u0026amp;nbsp;Our numerical algorithm reformulates the (N-1) strongly-coupled\u0026amp;nbsp;phase field equations for general order parameters\u0026amp;nbsp;into 2(N-1)\u0026amp;nbsp;Helmholtz-type equations that are completely de-coupled\u0026amp;nbsp;from one another. This leads to a computational complexity\u0026amp;nbsp;comparable to\u0026amp;nbsp;that for the simplified phase field equations\u0026amp;nbsp;associated with certain special choice of the order parameters.\u0026amp;nbsp;We demonstrate the capabilities of the method\u0026amp;nbsp;developed herein using several test problems involving multiple fluid phases\u0026amp;nbsp;and large\u0026amp;nbsp;contrasts in densities and viscosities\u0026amp;nbsp;among\u0026amp;nbsp;the multitude of fluids. In particular, by comparing\u0026amp;nbsp;simulation results with the Langmuir-de Gennes\u0026amp;nbsp;theory of floating liquid\u0026amp;nbsp;lenses we show that the method using general\u0026amp;nbsp;order parameters\u0026amp;nbsp;produces physically accurate results for multiple fluid phases.\u0026amp;nbsp;\u0026lt;/p\u0026gt;","descriptionType":"Abstract"},{"description":"We present a family of thermodynamically consistent physical formulation and efficient numerical algorithm for simulating the mixture of N (N \u0026gt;= 2) immiscible incompressible fluids with given densities, dynamic viscosities and...","descriptionType":"Other"}],"geoLocations":[],"fundingReferences":[{"awardTitle":"Joint Diagonalization-Based Spectral Element Approach","funderName":"NSF"}],"xml":"PD94bWwgdmVyc2lvbj0iMS4wIiBlbmNvZGluZz0iVVRGLTgiPz4KPHJlc291cmNlIHhtbG5zOnhzaT0iaHR0cDovL3d3dy53My5vcmcvMjAwMS9YTUxTY2hlbWEtaW5zdGFuY2UiIHhtbG5zPSJodHRwOi8vZGF0YWNpdGUub3JnL3NjaGVtYS9rZXJuZWwtNCIgeHNpOnNjaGVtYUxvY2F0aW9uPSJodHRwOi8vZGF0YWNpdGUub3JnL3NjaGVtYS9rZXJuZWwtNCBodHRwczovL3NjaGVtYS5kYXRhY2l0ZS5vcmcvbWV0YS9rZXJuZWwtNC41L21ldGFkYXRhLnhzZCI+CiAgPGlkZW50aWZpZXIgaWRlbnRpZmllclR5cGU9IkRPSSI+MTAuNDIzMS9SN0o5NjQ5UDwvaWRlbnRpZmllcj4KICA8Y3JlYXRvcnM+CiAgICA8Y3JlYXRvcj4KICAgICAgPGNyZWF0b3JOYW1lPkRvbmcsIFN1Y2h1YW48L2NyZWF0b3JOYW1lPgogICAgICA8Z2l2ZW5OYW1lPlN1Y2h1YW48L2dpdmVuTmFtZT4KICAgICAgPGZhbWlseU5hbWU+RG9uZzwvZmFtaWx5TmFtZT4KICAgICAgPG5hbWVJZGVudGlmaWVyIG5hbWVJZGVudGlmaWVyU2NoZW1lPSJPUkNJRCIgc2NoZW1lVVJJPSJodHRwczovL29yY2lkLm9yZy8iPjAwMDAtMDAwMS02Nzc4LTA2Nzk8L25hbWVJZGVudGlmaWVyPgogICAgICA8YWZmaWxpYXRpb24+UHVyZHVlIFVuaXZlcnNpdHk8L2FmZmlsaWF0aW9uPgogICAgPC9jcmVhdG9yPgogIDwvY3JlYXRvcnM+CiAgPHRpdGxlcz4KICAgIDx0aXRsZT5JbmNvbXByZXNzaWJsZSBNdWx0aXBoYXNlIEZsb3dzOiBQaHlzaWNhbCBGb3JtdWxhdGlvbiBhbmQgTnVtZXJpY2FsIEFsZ29yaXRobTwvdGl0bGU+CiAgPC90aXRsZXM+CiAgPHB1Ymxpc2hlciB4bWw6bGFuZz0iZW4iIHB1Ymxpc2hlcklkZW50aWZpZXI9Imh0dHBzOi8vd3d3LnJlM2RhdGEub3JnL3JlcG9zaXRvcnkvcjNkMTAwMDEwOTUxIiBwdWJsaXNoZXJJZGVudGlmaWVyU2NoZW1lPSJyZTNkYXRhIiBzY2hlbWVVUkk9Imh0dHBzOi8vd3d3LnJlM2RhdGEub3JnLyI+UHVyZHVlIFVuaXZlcnNpdHkgUmVzZWFyY2ggUmVwb3NpdG9yeTwvcHVibGlzaGVyPgogIDxwdWJsaWNhdGlvblllYXI+MjAxNTwvcHVibGljYXRpb25ZZWFyPgogIDxkYXRlcz4KICAgIDxkYXRlIGRhdGVUeXBlPSJBdmFpbGFibGUiPjIwMTUtMDQtMDc8L2RhdGU+CiAgICA8ZGF0ZSBkYXRlVHlwZT0iU3VibWl0dGVkIj4yMDE1LTA0LTA3PC9kYXRlPgogICAgPGRhdGUgZGF0ZVR5cGU9IkFjY2VwdGVkIj4yMDE1LTA0LTA3PC9kYXRlPgogIDwvZGF0ZXM+CiAgPGxhbmd1YWdlPmVuPC9sYW5ndWFnZT4KICA8cmVzb3VyY2VUeXBlIHJlc291cmNlVHlwZUdlbmVyYWw9IkRhdGFzZXQiPkRhdGFzZXRzPC9yZXNvdXJjZVR5cGU+CiAgPHJlbGF0ZWRJZGVudGlmaWVycz4KICAgIDxyZWxhdGVkSWRlbnRpZmllciByZWxhdGVkSWRlbnRpZmllclR5cGU9IkRPSSIgcmVsYXRpb25UeXBlPSJSZWZlcmVuY2VzIiByZXNvdXJjZVR5cGVHZW5lcmFsPSJKb3VybmFsQXJ0aWNsZSI+MTAuMTAxNi9qLmpjcC4yMDE0LjExLjAzOTwvcmVsYXRlZElkZW50aWZpZXI+CiAgPC9yZWxhdGVkSWRlbnRpZmllcnM+CiAgPHZlcnNpb24+MS4wPC92ZXJzaW9uPgogIDxyaWdodHNMaXN0PgogICAgPHJpZ2h0cyByaWdodHNVUkk9Imh0dHA6Ly9jcmVhdGl2ZWNvbW1vbnMub3JnL2xpY2Vuc2VzL2J5LW5jLW5kLzMuMC9sZWdhbGNvZGUiPkF0dHJpYnV0aW9uLU5vbkNvbW1lcmNpYWwtTm8gRGVyaXZzIDMuMCBVbnBvcnRlZDwvcmlnaHRzPgogIDwvcmlnaHRzTGlzdD4KICA8ZnVuZGluZ1JlZmVyZW5jZXM+CiAgICA8ZnVuZGluZ1JlZmVyZW5jZT4KICAgICAgPGZ1bmRlck5hbWU+TlNGPC9mdW5kZXJOYW1lPgogICAgICA8YXdhcmRUaXRsZT5Kb2ludCBEaWFnb25hbGl6YXRpb24tQmFzZWQgU3BlY3RyYWwgRWxlbWVudCBBcHByb2FjaDwvYXdhcmRUaXRsZT4KICAgIDwvZnVuZGluZ1JlZmVyZW5jZT4KICA8L2Z1bmRpbmdSZWZlcmVuY2VzPgogIDxmb3JtYXRzPgogICAgPGZvcm1hdD5pbWFnZS9qcGVnPC9mb3JtYXQ+CiAgICA8Zm9ybWF0PmFwcGxpY2F0aW9uL3BkZjwvZm9ybWF0PgogIDwvZm9ybWF0cz4KICA8c2l6ZXM+CiAgICA8c2l6ZT4xOCBmaWxlczwvc2l6ZT4KICAgIDxzaXplPjkgTUI8L3NpemU+CiAgPC9zaXplcz4KICA8c3ViamVjdHM+CiAgICA8c3ViamVjdD5NYXRoZW1hdGljczwvc3ViamVjdD4KICAgIDxzdWJqZWN0Pk4tcGhhc2UgZmxvdzwvc3ViamVjdD4KICAgIDxzdWJqZWN0PnBoYXNlIGZpZWxkPC9zdWJqZWN0PgogICAgPHN1YmplY3Q+bXVsdGlwaGFzZSBmbG93czwvc3ViamVjdD4KICAgIDxzdWJqZWN0PnN1cmZhY2UgdGVuc2lvbjwvc3ViamVjdD4KICAgIDxzdWJqZWN0PnNwZWN0cmFsIGVsZW1lbnQ8L3N1YmplY3Q+CiAgICA8c3ViamVjdD5nZW5lcmFsIG9yZGVyIHBhcmFtZXRlcnM8L3N1YmplY3Q+CiAgICA8c3ViamVjdD5wYWlyd2lzZSBzdXJmYWNlIHRlbnNpb248L3N1YmplY3Q+CiAgICA8c3ViamVjdD5hcHBsaWVkIG1hdGhlbWF0aWNzPC9zdWJqZWN0PgogICAgPHN1YmplY3Q+Y29tcHV0YXRpb25hbCBwaHlzaWNzPC9zdWJqZWN0PgogICAgPHN1YmplY3Q+UGh5c2ljczwvc3ViamVjdD4KICA8L3N1YmplY3RzPgogIDxkZXNjcmlwdGlvbnM+CiAgICA8ZGVzY3JpcHRpb24gZGVzY3JpcHRpb25UeXBlPSJBYnN0cmFjdCI+JmFtcDtsdDtwJmFtcDtndDtXZSBwcmVzZW50IGEgZmFtaWx5IG9mIHBoeXNpY2FsIGZvcm11bGF0aW9ucywmYW1wO2FtcDtuYnNwO2FuZCBhJmFtcDthbXA7bmJzcDsgbnVtZXJpY2FsIGFsZ29yaXRobSwmYW1wO2FtcDtuYnNwO2Jhc2VkIG9uIGEgY2xhc3Mgb2YgZ2VuZXJhbCBvcmRlciBwYXJhbWV0ZXJzIGZvciBzaW11bGF0aW5nJmFtcDthbXA7bmJzcDt0aGUgbW90aW9uIG9mIGEgbWl4dHVyZSBvZiBOJmFtcDthbXA7bmJzcDsoTiAmYW1wO2FtcDtndDs9JmFtcDthbXA7bmJzcDsyKSZhbXA7YW1wO25ic3A7aW1taXNjaWJsZSBpbmNvbXByZXNzaWJsZSBmbHVpZHMmYW1wO2FtcDtuYnNwO3dpdGggZ2l2ZW4gZGVuc2l0aWVzLCBkeW5hbWljIHZpc2Nvc2l0aWVzLCBhbmQgcGFpcndpc2UmYW1wO2FtcDtuYnNwO3N1cmZhY2UgdGVuc2lvbnMuJmFtcDthbXA7bmJzcDtUaGUgTi1waGFzZSBmb3JtdWxhdGlvbnMgc3RlbSBmcm9tIGEgcGhhc2UgZmllbGQgbW9kZWwmYW1wO2FtcDtuYnNwO3dlIGRldmVsb3BlZCBpbiBhIHJlY2VudCB3b3JrIGJhc2VkIG9uJmFtcDthbXA7bmJzcDt0aGUgY29uc2VydmF0aW9ucyBvZiBtYXNzL21vbWVudHVtLCBhbmQgdGhlJmFtcDthbXA7bmJzcDtzZWNvbmQgbGF3IG9mIHRoZXJtb2R5bmFtaWNzLiZhbXA7YW1wO25ic3A7VGhlIGludHJvZHVjdGlvbiBvZiBnZW5lcmFsIG9yZGVyJmFtcDthbXA7bmJzcDtwYXJhbWV0ZXJzIGxlYWRzIHRvIGFuIGV4dHJlbWVseSBzdHJvbmdseS1jb3VwbGVkIHN5c3RlbSZhbXA7YW1wO25ic3A7b2YgKE4tMSkmYW1wO2FtcDtuYnNwO3BoYXNlIGZpZWxkIGVxdWF0aW9ucy4mYW1wO2FtcDtuYnNwO09uIHRoZSBvdGhlciBoYW5kLCB0aGUgZ2VuZXJhbCBmb3JtIGVuYWJsZXMgb25lIHRvIGNvbXB1dGUgdGhlIE4tcGhhc2UmYW1wO2FtcDtuYnNwO21peGluZyBlbmVyZ3kgZGVuc2l0eSBjb2VmZmljaWVudHMgaW4mYW1wO2FtcDtuYnNwO2FuIGV4cGxpY2l0JmFtcDthbXA7bmJzcDtmYXNoaW9uIGluIHRlcm1zIG9mJmFtcDthbXA7bmJzcDt0aGUgcGFpcndpc2Ugc3VyZmFjZSB0ZW5zaW9ucy4mYW1wO2FtcDtuYnNwO1dlIHNob3cgdGhhdCB0aGUgaW5jcmVhc2VkIGNvbXBsZXhpdHkgaW4gdGhlIGZvcm0gb2YmYW1wO2FtcDtuYnNwO3RoZSZhbXA7YW1wO25ic3A7cGhhc2UgZmllbGQgZXF1YXRpb25zJmFtcDthbXA7bmJzcDthc3NvY2lhdGVkIHdpdGggZ2VuZXJhbCBvcmRlciBwYXJhbWV0ZXJzJmFtcDthbXA7bmJzcDtpbiBhY3R1YWxpdHkmYW1wO2FtcDtuYnNwO2RvZXMgbm90IGNhdXNlJmFtcDthbXA7bmJzcDtlc3NlbnRpYWwgY29tcHV0YXRpb25hbCBkaWZmaWN1bHRpZXMuJmFtcDthbXA7bmJzcDtPdXIgbnVtZXJpY2FsIGFsZ29yaXRobSByZWZvcm11bGF0ZXMgdGhlIChOLTEpIHN0cm9uZ2x5LWNvdXBsZWQmYW1wO2FtcDtuYnNwO3BoYXNlIGZpZWxkIGVxdWF0aW9ucyBmb3IgZ2VuZXJhbCBvcmRlciBwYXJhbWV0ZXJzJmFtcDthbXA7bmJzcDtpbnRvIDIoTi0xKSZhbXA7YW1wO25ic3A7SGVsbWhvbHR6LXR5cGUgZXF1YXRpb25zIHRoYXQgYXJlIGNvbXBsZXRlbHkgZGUtY291cGxlZCZhbXA7YW1wO25ic3A7ZnJvbSBvbmUgYW5vdGhlci4gVGhpcyBsZWFkcyB0byBhIGNvbXB1dGF0aW9uYWwgY29tcGxleGl0eSZhbXA7YW1wO25ic3A7Y29tcGFyYWJsZSB0byZhbXA7YW1wO25ic3A7dGhhdCBmb3IgdGhlIHNpbXBsaWZpZWQgcGhhc2UgZmllbGQgZXF1YXRpb25zJmFtcDthbXA7bmJzcDthc3NvY2lhdGVkIHdpdGggY2VydGFpbiBzcGVjaWFsIGNob2ljZSBvZiB0aGUgb3JkZXIgcGFyYW1ldGVycy4mYW1wO2FtcDtuYnNwO1dlIGRlbW9uc3RyYXRlIHRoZSBjYXBhYmlsaXRpZXMgb2YgdGhlIG1ldGhvZCZhbXA7YW1wO25ic3A7ZGV2ZWxvcGVkIGhlcmVpbiB1c2luZyBzZXZlcmFsIHRlc3QgcHJvYmxlbXMgaW52b2x2aW5nIG11bHRpcGxlIGZsdWlkIHBoYXNlcyZhbXA7YW1wO25ic3A7YW5kIGxhcmdlJmFtcDthbXA7bmJzcDtjb250cmFzdHMgaW4gZGVuc2l0aWVzIGFuZCB2aXNjb3NpdGllcyZhbXA7YW1wO25ic3A7YW1vbmcmYW1wO2FtcDtuYnNwO3RoZSBtdWx0aXR1ZGUgb2YgZmx1aWRzLiBJbiBwYXJ0aWN1bGFyLCBieSBjb21wYXJpbmcmYW1wO2FtcDtuYnNwO3NpbXVsYXRpb24gcmVzdWx0cyB3aXRoIHRoZSBMYW5nbXVpci1kZSBHZW5uZXMmYW1wO2FtcDtuYnNwO3RoZW9yeSBvZiBmbG9hdGluZyBsaXF1aWQmYW1wO2FtcDtuYnNwO2xlbnNlcyB3ZSBzaG93IHRoYXQgdGhlIG1ldGhvZCB1c2luZyBnZW5lcmFsJmFtcDthbXA7bmJzcDtvcmRlciBwYXJhbWV0ZXJzJmFtcDthbXA7bmJzcDtwcm9kdWNlcyBwaHlzaWNhbGx5IGFjY3VyYXRlIHJlc3VsdHMgZm9yIG11bHRpcGxlIGZsdWlkIHBoYXNlcy4mYW1wO2FtcDtuYnNwOyZhbXA7bHQ7L3AmYW1wO2d0OzwvZGVzY3JpcHRpb24+CiAgICA8ZGVzY3JpcHRpb24gZGVzY3JpcHRpb25UeXBlPSJPdGhlciI+V2UgcHJlc2VudCBhIGZhbWlseSBvZiB0aGVybW9keW5hbWljYWxseSBjb25zaXN0ZW50IHBoeXNpY2FsIGZvcm11bGF0aW9uIGFuZCBlZmZpY2llbnQgbnVtZXJpY2FsIGFsZ29yaXRobSBmb3Igc2ltdWxhdGluZyB0aGUgbWl4dHVyZSBvZiBOIChOICZhbXA7Z3Q7PSAyKSBpbW1pc2NpYmxlIGluY29tcHJlc3NpYmxlIGZsdWlkcyB3aXRoIGdpdmVuIGRlbnNpdGllcywgZHluYW1pYyB2aXNjb3NpdGllcyBhbmQuLi48L2Rlc2NyaXB0aW9uPgogIDwvZGVzY3JpcHRpb25zPgo8L3Jlc291cmNlPg==","url":"https://purr.purdue.edu/publications/1840/1","contentUrl":null,"metadataVersion":8,"schemaVersion":"http://datacite.org/schema/kernel-4","source":"mds","isActive":true,"state":"findable","reason":null,"viewCount":0,"viewsOverTime":[],"downloadCount":0,"downloadsOverTime":[],"referenceCount":1,"citationCount":0,"citationsOverTime":[],"partCount":0,"partOfCount":0,"versionCount":0,"versionOfCount":0,"created":"2015-04-07T01:00:36.000Z","registered":"2015-04-07T01:00:38.000Z","published":"2015","updated":"2025-12-18T20:11:57.000Z"},"relationships":{"client":{"data":{"id":"purdue.purduelib","type":"clients"}},"provider":{"data":{"id":"purdue","type":"providers"}},"media":{"data":{"id":"10.4231/r7j9649p","type":"media"}},"references":{"data":[{"id":"10.1016/j.jcp.2014.11.039","type":"dois"}]},"citations":{"data":[]},"parts":{"data":[]},"partOf":{"data":[]},"versions":{"data":[]},"versionOf":{"data":[]}}}}